The descriptive complexity of connectedness in Polish spaces

Gabriel Debs, Jean Saint Raymond, Jean Saint Raymond Fundamenta Mathematicae MSC: Primary 03E15, 28A05, 54D05; Secondary 54H05. DOI: 10.4064/fm754-7-2019 Published online: 20 December 2019

Abstract

We investigate the descriptive complexity of connectedness (and also pathwise connectedness and local connectedness) of Polish spaces, and prove that even in the framework of finite-dimensional euclidean spaces this complexity can be the highest possible, and much beyond the first projective classes $\boldsymbol{\Sigma}^1_1 $ and $\boldsymbol{\Pi}^1_1 $. In particular we prove that several of these notions are $\boldsymbol{\Pi}^1_2 $-complete.

Authors

  • Gabriel DebsSorbonne Université
    Université Paris Diderot, CNRS
    Institut de Mathématiques de Jussieu – Paris Rive Gauche, IMJ-PRG
    4 place Jussieu
    F-75252 Paris, France
    and
    Université Le Havre Normandie
    Institut Universitaire de Technologie
    Rue Boris Vian, BP 4006
    76610 Le Havre, France
    e-mail
  • Jean Saint Raymond
  • Jean Saint RaymondSorbonne Université
    Université Paris Diderot, CNRS
    Institut de Mathématiques de Jussieu – Paris Rive Gauche, IMJ-PRG
    4 place Jussieu
    F-75252 Paris, France
    e-mail

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